Some remarks on GL

In this chapter we state some results on the representation theory of
GL_{n}(Fq), without proofs, with the intention of motivating further reading.
The construction of representations of GL_{n}(Fq) follows that same principles
as in the case of GL_{2}(Fq). Parabolic induction (of which the constructions in
Chapter 2 are examples) is used to construct a large number of irreducible
representations of GL_{n}(Fq) from representations of GL_{m}(Fq), when m < n.
The parameterisation of such representations is, in some sense, related
to the representation theory of symmetric groups. The remaining
representations are called cuspidal and are parameterised by the Galois
orbits of primitive characters of F_{qn}^{*}. The irreducible representations come
in families, which reflect the parametrisation of conjugacy classes on
GL_{n}(Fq).

The process of parabolic induction is best thought of in terms of a graded
associative algebra. Let R_{n} denote the free abelian group generated by the
set of isomorphism classes of irreducible representations of GL_{n}(Fq). Set
R = ⊕_{n=1}^{∞}R_{n}. Let P_{n,n′} denote the subgroup of GL_{n+n′}(Fq) consisting of
matrices with block form

The cuspidal representations of GL_{n}(Fq) are those which are disjoint from
all representations of the form π′∘ π′′, where π′ and π′′ are irreducible
representations of GL_{n′}(Fq) and GL_{n′′}(Fq), where n = n′ + n′′ and n′ and n′′
are both positive.

Together with the ‘∘’ operation, cuspidal representations generate all of R.

The cuspidal representations of GL_{n}(Fq) have a nice parametrisation. A
character ω of F_{qn}^{*} is called primitive if there does not exists any d∣n such
that ω = N ∘ χ for any character χ of F_{qd}^{*}. Here N denotes the norm map
F_{qn} → F_{qd} (see Section B.3). The Galois group of F_{qn} over Fq acts on the set
of primitive characters of F_{qn}: ω^{g}(x) = ω(^{g}x) for an element g of the Galois
group, for each x F_{qn}.

Theorem 4.1. There is a canonical bijective correspondence
between the set of Galois orbits of primitive characters of F_{qn}^{*} and
isomorphism classes irreducible cuspidal representations of GL_{n}(Fq).

It should be noted that the number such orbits is the same as the
number of irreducible monic polynomials of degree n with coefficients in
Fq. These correspond precisely to the conjugacy classes of matrices
in GL_{n}(Fq) with irreducible characteristic polynomial. Moreover,
this correspondence has a nice manifestation in terms of character
values.

Theorem 4.2. Let f(t) is an irreducible monic polynomial of degree
n with coefficients in Fq with roots z_{1},…,z_{n} in F_{qn}, and let ω be a
primitive character of F_{qn}^{*}. Let π_{ω} denote the irreducible cuspidal
representation of GL_{n}(Fq) corresponding to the Galois orbit of ω.
Then

The primary decomposition for matrices (Corollary A.13) has an
analogy for representations of GL_{n}(Fq). Fix an irreducible cuspidal
representation π of some GL_{n}(Fq). Say that a representation ρ of GL_{m}(Fq) is
π-primary if it is a subrepresentation of some polynomial expression of
π in R. If ρ_{1},…,ρ_{n} are irreducible primary representations, with ρ_{i}
begin π_{i}-primary, where π_{1},…,π_{n} are pairwise non-isomorphic cuspidal
representations, then ρ_{1} ∘∘ ρ_{n} is irreducible.

Green shows that the irreducible π-primary representations are parameterised by partitions. It is no coincidence that the irreducible representations of symmetric groups are also parameterised by partitions. An elegant approach to understanding these relationships is by putting additional structure on R, namely that of a positive self adjoint Hopf algebra. Very general results about the structure of such algebras are interpreted in terms of the representation theory of general linear groups over finite fields by Zelevinsky in [Zel81].